# Simple Pendulum

In this notebook, we're solving the simple pendulum problem analytically and numerically. The simple pendulum is a classical example of a nonlinear, second-order, ordinary differential equation (ODE).

## Problem Statement

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The simple pendulum is a mass $m$ attached to a massless, frictionless rod of length $l$ that is free to rotate about a fixed point. The pendulum is released from rest at an angle $\theta_0$ from the vertical

## Necessary libraries

* `numpy` - for numerical calculations

* `matplotlib` - for plotting

* `scipy.integrate` - for solving ODEs

* `sympy` - for symbolic calculations

Run the next cell in case you don't have these libraries installed.

## Definitions

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We need to define the following constants and variables:

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* $g$ - acceleration due to gravity

* $l$ - length of the pendulum

* $m$ - mass of the pendulum

* $\theta$ - angle of the pendulum from the vertical (function of time)

* $t$ - time

Now we need to define $x$ and $y$ in terms of $\theta$.

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* $x$ - horizontal position of the pendulum - $x = l \sin(\theta)$

* $y$ - vertical position of the pendulum - $y = -l \cos(\theta)$

Once we have $x$ and $y$, we can get the kinetic and potential energies of the pendulum.

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* $T$ - kinetic energy of the pendulum 

    $$T = \frac{1}{2} m \dot{x}^2 + \frac{1}{2} m \dot{y}^2$$

* $V$ - potential energy of the pendulum

    $$V = m g y$$

Now, we can get the Lagrangian of the pendulum.

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* $L$ - Lagrangian of the pendulum - $L = T - V$

With the Lagrangian, we can get the equations of motion.

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$$\frac{d}{dt} \frac{\partial L}{\partial \dot{\theta}} - \frac{\partial L}{\partial \theta} = 0$$

Now, let's solve the equation for $\ddot{\theta}$.

With the analitycal solution for $\ddot{\theta}$, we can get the numerical solution for $\dot{\theta}$ and $\theta$.

Let's define a vector $\vec{S}$ that returns $\frac{d\vec{S}}{dt}$.

Now, let's plot the results.

Plot phase space using streamplots.

To create an animation, we need to define a function that returns $x$ and $y$ coordinates for each frame.